Direct, Indirect, and Self-Trapped Excitons in Cs2AgBiBr6

Cs2AgBiBr6 exhibits promising photovoltaic and light-emitting properties, making it a candidate for next-generation solar cells and LED technologies. Additionally, it serves as a model system within the family of halide double perovskites, offering insights into a broader class of materials. Here, we study various possible excited states of this material to understand its absorption and emission properties. We use time-dependent density functional theory (TD-DFT) coupled with nonempirical hybrid functionals, specifically PBE0(α) and dielectric-dependent hybrids (DDH) to explore direct, indirect, and self-trapped excitons in this material. Based on comparison with experiment, we show that these methods can give excellent predictions of the absorption spectrum and that the fundamental band gap has been underestimated in previous computational studies. We connect the experimental photoluminescence signals at 1.9–2.0 eV to the emission from self-trapped excitons and electron polarons. Finally, we reveal a complex landscape with energetically competing direct, indirect, and self-trapped excitons in the material.

H alide perovskites (HPs), materials with the general ABX 3 formula, have attracted significant interest for their advanced optoelectronic applications, including solar panels and lighting.A subclass within the HP family, halide double perovskites, replaces the single B metal cation with a combination of two different elements, yielding the A 2 BB′X 6 formula.Also known as elpasolites, these materials hold the potential for applications in photovoltaics, X-ray detection, and white light emission.Among halide double perovskites, Cs 2 AgBiBr 6 stands out as one of the most studied.It has gained interest as a viable alternative to lead-based perovskites due to its high stability, nontoxicity, outstanding optoelectronic properties, and multifunctionality.Additionally, it has emerged as a benchmark case for both experimental and computational studies.
One of the interesting properties of halide double perovskites is that they often exhibit significant light emission.In Cs 2 AgBiBr 6 , a photoluminescence (PL) peak has been measured at 1.9−2.0−3 This emission has been assigned to the indirect band gap 2,4,5 or subgap states 6 such as colorcenters. 1,3It is therefore useful to computationally assess the likelihood of different sources of emission.Since the possible interpretations involve both free (in particular indirect) excitons and self-trapping, such a comparison requires a method that can model energetics of delocalized and localized excitations on the same footing.
−11 However, the high computational cost of BSE makes its use impractical in the case of self-trapped excitons (STEs), which in the most commonly used approach require the use of supercells.One notable exception is a study by Ismail-Beigi et al. 12 on the STE in SiO 2 .We note that recent developments by Dai et al. 13,14 made it possible to model STEs in unit cells.−17 Alternatively, recent advancements have allowed for more computationally efficient alternatives without compromising the accuracy inherent in BSE.Most notably, the combination of time-dependent density functional theory (TD-DFT) with nonempirical hybrid functionals has been shown to achieve the accuracy of BSE at a fraction of the cost. 18,19While this method so far has been used for pristine materials and free excitons, its predictive power and relatively high computational efficiency make it a promising method for studying STEs as well.In a recent study by Jin et al. 20 it has been shown that STEs can be efficiently studied using the TD-DFT method.At the same time, they showed that the constrained-occupation DFT method, also called ΔSCF, leads to very similar geometries using TD-DFT forces to relax the structure of the STE.
In the present study, we combine TD-DFT with two types of nonempirical hybrid functionals.−23 Another one belongs to the class of dielectricdependent hybrid functionals (DDH), 24−26 in which the exchange potential follows the inverse of the dielectric function.From these methods, we extract the transitions of the direct and indirect free excitons as well as of the selftrapped exciton in the singlet and triplet state.We show that different excited states can be close in energy, and by performing careful convergence studies, we obtain the emission energy from the singlet state of the STE close to the experimental PL signal.
Calculations are performed using two simulation cells.Parametrization of the DDH functional, most convergence tests, and the comparison with the experimental absorption spectra are based on the unit cell of Cs 2 AgBiBr 6 (space group Fm3̅ m) using the experimental lattice parameter of 11.27 Å. 28 Calculations for the free excitons and the STE are done in a supercell containing 320 atoms, based on the polymorphous cubic structure 16 from ref 23, where we used the ΔSCF method for the excited state.In that work, we studied charge localization in Cs 2 AgBiBr 6 and showed that small electron polarons and STEs are stable in the material, while the localization of small hole polarons is less favorable.We note that these results align well with the later findings by Lafuente-Bartolome et al. 29 From ref 23, we adopt the α parameter in PBE0(α) of 0.28, which has been shown to satisfy the Koopmans' condition for the Br vacancy.All simulations presented in the main text include spin−orbit coupling (SOC), while some of the convergence tests shown in the Supporting Information are done without this contribution.Additional computational details and convergence tests can be found in the Supporting Information.
We first calculate the dielectric function ϵ of Cs 2 AgBiBr 6 within the random-phase approximation (RPA).The calculation is carried out using an energy cutoff of 400 eV and a kpoint mesh of 6 × 6 × 6.As shown in the Supporting Information, these parameters lead to a very well converged dielectric function.The result is given in Figure 1a in which we also include the fit following where ϵ lr −1 corresponds to the long-range exchange fraction at G = 0, G is the wave vector, and μ is the range-separation parameter.We obtain ϵ lr −1 = 0.19 and μ = 1.1.To verify the validity of our hybrid functionals, we compare the absorption spectra calculated via TD-DFT with experimental results from ref 3 (see Figure 1b).The absorption α abs is calculated based on the real and imaginary parts of the dielectric function ϵ as where c is the speed of light and ω is the angular frequency. 30onvergence tests corresponding to the TD-DFT calculations are given in the Supporting Information.We observe a good agreement with the experimental results using both hybrid functionals, with TD-PBE0(28%) reproducing the position of the first absorption peak almost exactly and TD-DDH slightly overestimating it, while giving a better agreement for the valley and the second peak.We attribute the overestimation of the second peak within TD-PBE0(28%) to its incorrect asymptotic behavior.Using the nonempirical hybrid functionals, we obtain a much better agreement with experimental data than previous G 0 W 0 calculations, 31 where the position of the first peak was underestimated by about 0.6 eV.This is due to the fact that while nonempirical hybrid functionals have been shown to give high accuracy in describing band gaps of halide perovskites, 22,32 the one-shot G 0 W 0 used in ref 31 can significantly underestimate that property. 15,33,34The authors of ref 31 noted that this underestimation is primarily due to the lack of selfconsistency and the starting point dependence of G 0 W 0 calculations, which is a well-known issue.Although other calculations in ref 5 considered partial self-consistency by updating energies in G and W (the evGW scheme), this approach still did not lead to the full increase of the band gap that would reproduce experimental results.In Table 1 we include the fundamental direct and indirect gaps calculated with the DDH and PBE0(α) methods and compare them with previous computationally reported values (the band structure  The Journal of Physical Chemistry Letters is given in the Supporting Information).First, we note that the two hybrid functionals constructed by using different physical considerations lead to almost the same indirect and direct band gaps.Our indirect band gap is also close to the value recently calculated by Wang et al. using the double screened hybrid (DSH) functional. 35Second, even when similar methods are employed, such as G 0 W 0 on top of LDA or PBE, band gaps can differ by as much as 0.35 eV.This discrepancy can be attributed to differences in the underlying functional, codes used, the choice between PAW and norm-conserving pseudopotentials, and the selection of valence states.Third, we observe that the values present here, about 2.7 and 3.5 eV for the indirect and direct transitions, respectively, are higher than most previous estimates.However, considering the excellent agreement between absorption spectra calculated here and the experiment, we argue that the presented fundamental gaps are more reliable predictions than the previously reported values.
The previous calculations were performed for the perfect cubic structure of Cs 2 AgBiBr 6 .−17 We note that ref 37 has shown that unlike lead-based perovskites, Cs 2 AgBiBr 6 has well-defined normal modes up to room temperature, which means that the symmetric average cubic structure is enough to describe the electronic structure of the material even at finite temperatures.We demonstrate this by comparing the imaginary part of the dielectric function calculated within the pristine Fm3̅ m unit cell and the polymorphous supercell in the Supporting Information.However, introducing local distortions within the perfect cubic supercell would lead to its relaxation to a form of a polymorphous cell or to finding a local minimum where the optimal charge localization is not possible. 23Therefore, we now turn to the study of the excited states of Cs 2 AgBiBr 6 based on a more realistic polymorphous model.We adopt one polymorphous structure for the pristine Cs 2 AgBiBr 6 and one for the STE from ref 23.We note that we have also tested how the properties of the STE change when the low-temparature tetragonal structure (I4/m) is considered and found that similar transitions can be found in that model (see Supporting Information for the test).These structures were generated using the CP2K code, 38,39 and here, for consistency, we further relax the atomic position within VASP.In the following, we focus on the PBE0(28%) functional, as it gives a slightly better agreement with experiment for the lower part of the absorption spectrum.The STE is relaxed within its lowest triplet state, and we assume that the atomic positions do not change significantly for the singlet state.We show the isodensities of the localized hole and electron in Figure 2a).We then perform TD-DFT calculations using the two hybrid functionals on top of these geometries.While the relaxation is performed using the Γ point only, for the TD-DFT calculations we use a special grid with four k-points.As shown in the Supporting Information, this grid yields lower parts of the absorption spectra that agree well with those from the 2 × 2 × 2 grid.The differences in the positions of the two lowest peaks between the two grids are below 0.03 eV.In the pristine material, we observe the lowest transition at 2.45 eV.This transition is dark and corresponds to the lowest indirect exciton.It would not be captured in the unit cell, unless momentum transfers are considered as in ref 5.However, in the supercell used in this study, the X and L points are folded onto the Γ point and are directly accessible by TD-DFT.From the difference between this dark transition and the fundamental band gap in the polymorphous supercell of 2.83 eV we estimate the binding energy of the lowest indirect free exciton to be 0.39 eV.This is slightly lower than the value of 0.48 eV, reported by Palummo et al. 5 The first bright transition, corresponding to the lowest direct free exciton is found at 2.96 eV, while the lowest direct independent-particle transition equals 3.57 eV in the same setup.This implies the binding energy of the direct exciton of 0.61 eV.Within the unit cell, we find the binding energy to amount to 0.55 eV (see the related discussion in Supporting Information).We note that this value is significantly higher

The Journal of Physical Chemistry Letters
than previously reported ones (0.34 eV in ref 5 and 0.17 eV in ref 31), which can be assigned to the higher fundamental band gap of Cs 2 AgBiBr 6 in the present setup.In their study, Biega et al. 31 demonstrated a linear relationship between the exciton binding energy and the size of the direct band gap, which would explain the higher value found here.In the Supporting Information we also demonstrate that by reducing the amount of exact exchange in the PBE0 functional, the exciton binding energy is reduced and in good agreement with previous studies in which the band gaps were also found to be smaller.
We now analyze the transitions in the TD-DFT calculations for the STE geometry.We note that the calculation corresponds to the excitation from the singlet ground state.However, since absorption and emission are inverse processes, the calculated energy transitions can be expected to be directly comparable to measured photoluminescence spectra.The calculated transitions of the STE are given in Figure 2b.While TD-DFT gives all transitions, we only plot those below 2 eV, as they correspond to the energy range in which photoluminescence is measured.We compare the results with the experimental PL spectrum from ref 3. The measurement was done at 4 K; hence, the phonon broadening can be neglected in the comparison.While the simulated spectrum is at lower energies than the experimental one, the difference between the midpoints of the spectra is only about 0.2 eV, representing fairly good agreement.We also note that the broadness of the experimental peak at low temperatures can be explained by the distribution of transitions that contribute to it.Previous works also suggested the possibility that the experimental emission is due to the indirect band gap 2,4,5 or subgap states 6 such as color-centers. 1,3We rule out emission from the indirect band gap based on two arguments.First, we find this transition at a significantly higher energy, 2.49 eV.Second, this type of recombination requires an additional momentum change to occur.As for defects, to contribute to the emission, they would need to be deep and trap charges more strongly than STE or small polarons.In that case, they would also lead to transitions at even lower energies than what we report here.In Figure 2c we show the band composition of the calculated transitions.The lowest transition at 1.42 eV has a very low intensity and corresponds to a spin-forbidden triplet-singlet transition between the localized electron and hole states from within the STE.Then, at 1.68 eV there is a bright transition from the singlet state of the STE.This implies a separation of 0.26 eV between the lowest triplet and higher singlet states of the STE in Cs 2 AgBiBr 6 .At higher energies, between 1.74 and 1.95 eV, there is a distribution of transitions which all involve the localized electron state and both localized and delocalized hole states.This range of energies can be used as an estimate of PL stemming from the electron polaron if the STE dissociates.In this energy range, we do not observe any transitions between the localized hole and delocalized electrons because the hole in the STE has a more shallow level than the electron, and this type of emission would mostly overlap with the absorption onset close to 3 eV.
Using the energy transitions from the TD-DFT calculations, we finally construct the configuration coordinate diagram.It allows us to compare the energies of direct, indirect, and selftrapped excitons.The diagram, given in Figure 3, is based on TD-PBE0(28%) calculations in the polymorphous cubic supercell.The energies at the ground-state and STE geometries are explicitly calculated and connected by parabolic curves to schematically represent the energy dependence on displacement.The diagram reveals a complex landscape of electron−hole excitations in the material and can be used to analyze the dynamics of the excited charges.First, direct free excitons are created.These free excitons can then be trapped in the form of the singlet STE.This STE can then undergo four processes.One, it can recombine, leading to a bright emission that can be measured in photoluminescence.Two, it can lower its energy and turn into a triplet form of STE.Three, it can detrap and become an indirect free exciton, which has an energy very close to that of the triplet STE (within 0.1 eV in the current computational setup).Finally, it could also dissociate leaving behind a localized electron and possibly a delocalized hole. 23We note that this diagram provides a more comprehensive picture of the energetics involved compared to ref 23, which only compared the energy of the triplet STE with the lowest indirect free-carrier transition.
In conclusion, we have studied different types of excitons in Cs 2 AgBiBr 6 halide double perovskites.We first assessed the performance of two nonempirical hybrid functionals, PBE0(28%) and DDH in the description of the optical properties of the pristine material.We have shown that both of them predict band gaps higher than what was previously reported in the literature and give absorption spectra in very good agreement with experiment.We then used the TD-PBE0(28%) technique to study excitations in the polymorphous cubic supercells corresponding to the ground-state structure and the self-trapped exciton.This allowed us to show that the emission from the STE is in good agreement with experimental PL spectra.Based on a configuration coordinate diagram, we finally revealed a complex landscape of electron− hole pairs in the materials with direct, indirect, and self-trapped excitons having comparable energies.

Data Availability Statement
Structures of the pristine material and of the self-trapped exciton are publicly available via Zenodo at https://doi.org/10.5281/zenodo.13258153.

Figure 1 .
Figure 1.(a) Inverse dielectric function versus wave vector at the Γ point calculated within the RPA and fitted with eq 1.(b) Absorption spectra calculated using TD-DDH and TD-PBE0(28%) compared with the experimental results from ref 27.Computed absorption spectra include a convolution with Lorentzians with a width of 0.07 eV and were renormalized to match the intensity of the first experimental peak.

Figure 2 .
Figure 2. (a) Isodensities of the hole (in red) and electron (in blue), localized within the AgBr 6 and BiBr 6 octahedron, respectively.The rest of the structure was removed for clarity.(b) Emission from the STE compared with the experimental PL spectrum from ref 3. The broadened spectrum was generated by convolution with Lorentzians with a width of 0.03 eV.All spectra were rescaled to have the same maximum.(c) Band contributions to each of the transitions marked in the upper panel.For each initial (final) state, the contributions of different final (initial) states, as well as all k-points, are added up.

Figure 3 .
Figure 3. Configuration coordinate diagram of different electron− hole pairs in Cs 2 AgBiBr 6, including free carriers (FC), free excitons (FE), and self-trapped excitons (STE)."Indir" and "Dir" refer to indirect and direct transitions, and "S" and "T" refer to singlet and triplet states.